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Abstract

We introduce the concept of adiabatic four-wave mixing (AFMW) frequency conversion in cubic nonlinear media through an analogy to dynamics in quantum two-level systems. Rapid adiabatic passage in four-wave mixing enables coherent near-100% photon number down-conversion or up-conversion over a bandwidth much larger than ordinary phase-matching bandwidths, overcoming the normal efficiency-bandwidth trade-off. We develop numerical methods to simulate AFWM pulse propagation in silicon photonics and fiber platforms as examples. First, we show that with a longitudinally varying silicon waveguide structure, a bandwidth of 70 nm centered at 1820 nm can be generated with 90% photon number conversion. Second, we predict the broadband generation of nanojoule energy, 4.2–5.2 μm mid-infrared light in a short, linearly tapered fluoride step-index fiber. We expect the AFWM concept to be broadly applicable to cubic nonlinear platforms, for applications as diverse as bright ultrafast light pulse generation, sensing, and conversion between telecommunications bands.

Figures (6)

Fig. 1 Energy diagrams of two schemes of four-wave mixing, and the analogy between two-level atoms and FWM. (a) Annihilation of Signal and Pump A photons allows creation of Idler and Pump B photons. This scheme is known as four-wave mixing Bragg scattering. (b) Annihilation of a Signal photon allows creation of Pump A, Pump B, and Idler photons. In both cases, when pump intensities are much larger than those of the signal and idler, signal and idler photons are exchanged through coupled equations of motion possessing SU(2) symmetry, analogously to population transfer in coupled two-level quantum systems. (c),(d) Analogy between Stark-chirped two-level atomic systems with field-free energies E1, E2 and four-wave mixing. ħΩP is the coupling pump’s photon energy in two-level atomic systems. β′is (i = A,B,Sig,Idl) are wave-vectors of the four waves in FWM. Energy detuning Δ0 corresponds to phase mismatch Δk, while Stark-induced energy shift ΔS corresponds to Kerr-induced nonlinear phase modulation ∊0χ(3)γint2(ωBPB−ωAPA). The effective energy detuning Δeff is analogous to effective phase mismatch Δkeff.

Fig. 2 Adiabatic frequency conversion in silicon photonics. (Left) The scheme allows an efficient broadband conversion of signal to idler waves (or vice-versa) by adiabatically changing the wave-vector mismatch through a chirped modulation of the width along the propagation direction. (Right) Momentum matching diagram (see Fig. 1(d)) illustrating the use of a variable effective momentum, Kg(z), to longitudinally sweep the wave-vector mismatch. We have shown a near 100% conversion efficiency of an incoming signal spanning from 1.65–1.72 μm to an idler spanning from 1.78–1.86 μm.

Fig. 3 Simulation results for an adiabatically modulated Si waveguide and for the standard periodic QPM approach. (a) Pseudocolor plot of idler intensity (arb. units) versus wavelength for the standard case of a periodic width modulation (i.e., constant-valued Kg) chosen to phase match only the central carrier wavelength. Efficiency is highly non-uniform and displays conversion-back-conversion cycles. (b) Pseudocolor plot of idler intensity (arb. units) versus wavelength for the adiabatic case, which shows nearly full and highly uniform conversion for all wavelengths included in the simulation. As expected, the locations where each wavelength has its rapid conversion jump are separated longitudinally. (c) The sum of effective phase mismatch and added momenta, Δkmod(z), a multi-valued function corresponding to the wide range of signal (and corresponding idler) wavelengths. The overall shift is due to the longitudinal variation of Kg(z), the effective momentum of the modulated waveguide, while the smaller oscillations are due to the modulation of the propagation constants due to the changing waveguide dispersion.

Fig. 4 (a) Integrated idler and signal powers during propagation in periodic and aperiodic (adiabatic) waveguides, illustrating signal-to-idler energy transfer. For the adiabatic evolution there is convergence of energy transfer from signal (blue) to idler (orange), while the conventional QPM case (gray, purple) shows unwanted back-conversion with efficiency that is non-uniform and highly sensitive to the initial conditions. (b) Efficiency vs. propagation length for various idler wavelengths in the adiabatic waveguide (corresponding to cross-sections of Fig. 3(a)). Illustrating evolution of an adiabatic nature, convergence is apparent for each wavelength, with rapid conversion taking place at a different location corresponding to the zero-crossing of the QPM wave-vector mismatch.

Fig. 5 The relative positions of the ZDW and optical wavelengths determine the likelihood of a broad phase-matching bandwidth for AFWM in SIF. Top row: Propagation constants of the four mixing waves where ωA + ωSig = ωB + ωIdl and 2ωA = ωB, for three cases. The y-axis lies at the ZDW. Achieving Δkeff = 0 requires βM1 = βM2, where βM1 = (βA + βSig)/2 and βM2 = (βB + βIdl)/2. Bottom row: wavevector mismatch versus fiber core radius and idler wavelength for laser parameters corresponding to each case. (a),(d) Case for a standard silica SIF where all four waves are on one side of the ZDW. No phase matching is possible. (b),(e) Case where one frequency is on the opposite side of the ZDW from the other three. Narrowband phase matching under limited conditions is possible. The black curve represents contour zero (Δkeff = 0). (c),(f) Case where there are two waves on each side of the ZDW. Phase matching is probable in this case, and gives rise to conditions where an octave-spanning idler wave covering 3.5–7 μm can be phase matched in an AFWM process in a tapered fiber.

Fig. 6 Mid-IR generation via AFWM in InF3 SIF with core radius tapered from 4.5 to 3.55 μm over a 5-cm length. (a) Mid-IR beam spectral density evolution along the propagation axis, given a Gaussian input signal spectrum. AFWM can be observed for 4.2–5.2 μm. (b) Evolution of PCR for selected wavelengths. (c) PCR at the fiber exit vs. idler wavelength. Due to parasitic FWM amplification of the input signal, PCR determined by solution of the GNLSE is higher than that predicted by the cw simulation and includes values greater than 1. (d) Comparison between the spectral phases of the output mid-IR and input signal waves indicates the coherence of the generated wave. The idler’s phase consists of a conversion phase due to the wavelength-dependent conversion position (which is largely an effective third-order dispersion) plus the initial linear chirp (parabolic phase) of the signal pulse.